Advanced Calculus Single Variable

7.5 Finding The Derivative

Obviously there need to be simple ways of finding the derivative when it exists. There are
rules of derivatives which make finding the derivative very easy. In the following theorem,
the derivative could refer to right or left derivatives as well as regular derivatives.

Note the last part is the usual definition of the derivative given in beginning calculus
courses. There is nothing wrong with doing it this way from the beginning for a function of
only one variable but it is not the right way to think of the derivative and does not generalize
to the case of functions of many variables where the definition given in terms of o

(h)

does.

Corollary 7.5.2Let f′

(t)

,g′

(t)

bothexist and g

(t)

≠0, then the quotient ruleholds.

( )
f-′ f′(t)g(t)−-f (t)g′(t)
g = g(t)2

Proof: This is left to you. Use the chain rule and the product rule. ■

Higher order derivatives are defined in the usual way.

f′′ ≡ (f′)′

etc. Also the Leibniz notation is defined by

dy-= f′(x) where y = f (x)
dx

and the second derivative is denoted as

2
d-y
dx2

with various other higher order derivatives defined in the usual way.

The chain rule has a particularly attractive form in Leibniz’s notation. Suppose y = g

(u )

and u = f

(x)

. Thus y = g ∘ f

(x )

. Then from the above theorem

(g∘f)

′

(x)

= g′

(f (x))

f′

(x)

= g′

(u)

f′

(x)

or in other words,

dy- dydu-
dx = dudx .

Notice how the du cancels. This particular form is a very useful crutch and is used extensively
in applications.